find factorisation of 1-(a^2+b^2)+2ab

find factorisation of 1-(a^2+b^2)+2ab

Factorise: 1 - (a² + b²) + 2ab

Step 1: Open the bracket

1 - (a² + b²) + 2ab

= 1 - a² - b² + 2ab

Step 2: Rearrange the terms

= 1 - (a² - 2ab + b²)

Why?

Because:

-(a² - 2ab + b²)

= -a² + 2ab - b²

So,

1 - a² - b² + 2ab

= 1 - (a² - 2ab + b²)

Step 3: Use the identity

a² - 2ab + b² = (a - b)²

Therefore,

= 1 - (a - b)²

Step 4: Apply the identity

x² - y² = (x - y)(x + y)

Here,

1 - (a - b)²

= [1 - (a - b)][1 + (a - b)]

Step 5: Simplify each factor

= (1 - a + b)(1 + a - b)

Hence,

1 - (a² + b²) + 2ab

= (1 - a + b)(1 + a - b)

Final Answer:

(1 - a + b)(1 + a - b)